文章基本信息

标题：Uniqueness and concentration for a fractional Kirchhoff problem with strong singularity
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作者：Shengbin Yu ; Jianqing Chen
期刊名称：Boundary Value Problems
印刷版ISSN：1687-2762
电子版ISSN：1687-2770
出版年度：2021
卷号：2021
期号：1
页码：1
DOI：10.1186/s13661-021-01507-8
出版社：Hindawi Publishing Corporation
摘要：In this paper, we consider the following fractional Kirchhoff problem with strong singularity: $$ \textstyle\begin{cases} (1 b\int _{\mathbb{R}^{3}}\int _{\mathbb{R}^{3}} \frac{ \vert u(x)-u(y) \vert ^{2}}{ \vert x-y \vert ^{3 2s}}\,\mathrm{d}x \,\mathrm{d}y )(-\Delta )^{s} u V(x)u = f(x)u^{-\gamma }, & x \in \mathbb{R}^{3}, \\ u>0,& x\in \mathbb{R}^{3}, \end{cases} $$ where $(-\Delta )^{s}$ is the fractional Laplacian with $00$ is a constant, and $\gamma >1$ . Since $\gamma >1$ , the energy functional is not well defined on the work space, which is quite different with the situation of $0<\gamma <1$ and can lead to some new difficulties. Under certain assumptions on V and f, we show the existence and uniqueness of a positive solution $u_{b}$ by using variational methods and the Nehari manifold method. We also give a convergence property of $u_{b}$ as $b\rightarrow 0$ , where b is regarded as a positive parameter.
关键词：35A15 ; 35R11 ; 35B09 ; 35J75